In a graph G, a vertex is said to dominate itself and all vertices adjacent to it. For a positive integer k, the k-tuple domination number γ×k(G) of G is the minimum size of a subset D of V (G) such that every vertex in G is dominated by at least k vertices in D. To generalize/improve known upper bounds for the k-tuple domination number, this paper establishes that for any positive integer k an...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...

a dominating set $d subseteq v$ of a graph $g = (v,e)$ is said to be a connected cototal dominating set if $langle d rangle$ is connected and $langle v-d rangle neq phi$, contains no isolated vertices. a connected cototal dominating set is said to be minimal if no proper subset of $d$ is connected cototal dominating set. the connected cototal domination number $gamma_{ccl}(g)$ of $g$ is the min...